- An ordinary sphere
is 2-sphere
in the n-sphere classification scheme
because its surface is a 2-dimensional curved space.
- The study of curved spaces
in general
is called differential geometry
or, virtually synonymously,
non-Euclidean geometry.
Actually,
Euclidean geometry (AKA flat geometry)
is considered a special case of
non-Euclidean geometry.
- The surface of sphere
is probably the simplest
curved space to understand
by human intuition.
We will explicate
spherical geometry a bit below.
- First note that the generalization of
a straight line
to non-Euclidean geometry
is called
geodesic.
It is the
stationary path
between points
in spaces with general curvature.
A stationary path is
one where infinitesimal variations from it cause
**NO**change in length. Minimum and maximum paths, (global or local) are stationary paths. Stationary paths are analyzed in variational calculus.For spherical geometry, the geodesics are great circles which are circles that cut a sphere in half. Circles that do

**NOT**cut a sphere in half are small circles.Airways for aviation often follow great circles on Earth at least approximately since that shortens travel distance and travel time. This is why flights from New York City to Paris often go over Greenland.

- Now two
geodesics
are parallel
in general
where a third geodesic
intersects them both at 90°.
But in general such geodesics
are only parallel
for certain places.
Consider the sphere in the image. Note that the equator intersects two meridians both at 90°, but they are

**NOT**parallel elsewhere and, in fact, meet at the poles.In Euclidean geometry, of course, geodesics (i.e., straight lines) that are parallel at one place are parallel everywhere and

**NEVER**meet. This is the same as saying that anywhere along them there is a third geodesic that intersects them both a 90° and has the same length. - The generalization of the triangle
to a general 2-dimensional
space is
three geodesic segments that
are joined at vertices.
The sum of the
vertex
angles (measured within the
2-dimensional space)
is
**NOT**in general 180°.By inspection of the image, it is clear that the sum of the vertex angles for a triangle in spherical geometry is greater than 180°.

Caption: An illustration of the
surface of a sphere which is
a 2-dimensional curved space.
Note it is a finite, but unbounded space: there
is **NO** boundary.
The geometry of the surface
is called
spherical geometry.

Features:

Image link: Itself.

Local file: local link: space_spherical.html.

File: Mathematics file: space_spherical.html.